Abstract

Under an electric field, spherical conducting particles in a dielectric liquid assemble into a dendritic tree in order to dissipate charge. Several topological measures characterize such networks, including degree distributions, Strahler numbers, and total external pathlengths. Here, scaling laws relating these measures to the number of nodes in the system are presented and shown to match diffusion limited aggregation (DLA) structures. Experimental bifurcation and stream-length ratios in this easily reproducible laboratory experiment are found to agree with DLA simulations and Horton’s laws for rivernetworks. Certain scaling relations in transportation networks are known to originate from general features of networks. Here we find the experimental structures share properties with hydrological drainage networks.

Received 29 September 2010Accepted 23 November 2010Published online 08 February 2011

Acknowledgments:

This material is based upon work supported by the National Science Foundation Grant Nos. NSF PHY 01-40179 and NSF DMS 03-25939 ITR. The authors gratefully acknowledge the support for this work by Defense Advanced Research Projects Agency (DARPA) Physical Intelligence subcontract (HRL9060-000706).